Minimal prime initials and zero-dimensional spaces

For Commutative semigroups, prime ideals minimal among all primes containing a fixed ideal were characterized in [6], where it was shown that families of such minimal primes. form zero-dimensional spaces when provided with the hull-kernel topology. Here, we consider the analogous condition for initial sets in preordered sets: it is sufficient for minimality of a prime initial set among all prime initial sets containing a fixed initial, but it is not necessary. Several examples are presented.